Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term

In this paper we study the initial-boundary value problem of the multidimensional viscoelasticity equation with nonlinear source term . By using the potential well method, we first prove the global existence. Then we prove that when time , the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.

This paper considers the initial-boundary value problem (IBVP) of the multidimensional viscoelasticity equation with nonlinear source term

where is the unknown function with respect to the spacial variable and the time variable is a bounded domain.

The viscoelasticity equation

was suggested and studied by Greenberg et al.[1,2] from viscoelasticity mechanics in 1968. Under the condition and higher smooth conditions on and the initial data, they obtained the global existence of classical solutions for the initial-boundary value problem of Eq. (1.4).

After that many authors [3-11] studied the global well-posedness of IBVP for Eq. (1.4). In [3-10] the global existence, uniqueness and stability of solution were studied thoroughly. And in [11] the blow up of solution was discussed. Furthermore, in [12-14] the global existence of solution for IBVP of some multidimensional viscoelasticity equation was considered. And in [11] the blow up of solution for IBVP of the multidimensional generalisation of Eq. (1.4) was proved. Recently, in [15] and [16], the IBVP of the multidimensional viscoelasticity equation with nonlinear damping and source terms

was studied, and by using the potential well method, the global existence of weak solution was proved under some assumptions on nonlinear functions , , and the initial data. But we do not know how the global solution behaves as the time goes to infinity, namely the asymptotic behaviour of problem (1.1)-(1.3) is still open up to now. In the present paper, we try to study this problem by the multiplier method [17-22].

The main purpose of present paper is to consider the asymptotic behaviour of solution for problem (1.1)-(1.3). Since in the proof of the asymptotic behaviour of solution the global existence theory is required, it is necessary to give the proof of global existence of solution for problem (1.1)-(1.3).

In this paper, suppose that and satisfy the following assumptions:

where the constants in (H₁) and (H₂) are all positive and satisfy

In this paper, we first give some definitions and lemmas (Section 2). Then we prove the global existence of solution (Section 3). Finally, we prove the asymptotic behaviour of solution (Section 4).

In this paper, we denote by , and .

In this section, we will give some definitions and prove some lemmas for problem (1.1)-(1.3).

For problem (1.1)-(1.3), we define

Remark 2.1 Note that the definitions of and in the present paper are different from those in [11] and [15]. The definitions given in this paper will be shown more natural and rational because they are a part of the total energy .

Lemma 2.2Let (H₁) and (H₂) hold. Set

Then the following hold:

(i) is increasing and;

(ii) .

Proof This lemma follows from and . □

Lemma 2.3Let (H₁) and (H₂) hold, . Then the following hold:

(i) If, then;

(ii) If, then;

(iii) If, then,

where

Proof

(i) If , then we have

which gives .

(ii) If , then we have

which gives

(iii) If and , then by

we get

 □

Lemma 2.4Let (H₁) and (H₂) hold. Then the following holds:

Proof For any , by Lemma 2.3, we have and

which gives (2.1). □

Now, for problem (1.1)-(1.3), we define

In this section, we prove the global existence of weak solution for problem (1.1)-(1.3).

Definition 3.1 We call a weak solution of problem (1.1)-(1.3) on if , satisfying

(i)

(ii)

Theorem 3.2Let (H₁) and (H₂) hold, , . Assume that, . Then problem (1.1)-(1.3) admits a global weak solutionand.

Proof Let be a system of base functions in . Construct the approximate solutions of problem (1.1)-(1.3)

satisfying

Multiplying (3.1) by and summing for s, we get

and

where

From (3.2) and (3.3), we have as . Hence, for sufficiently large n, we have and

On the other hand, since W is an open set in , Eq. (3.2) implies that for sufficiently large n, we have . Next, we prove that for and sufficiently large n. If it is false, then there exists a such that , i.e. and , i.e.. So, by the definition of d, we get , which contradicts (3.6).

From (3.6) we have

which gives

and

which together with gives

and

From (3.7) we can get

Hence there exist u, , η and a subsequence of such that as , in weak-star, and a.e. in , in weak-star and in weakly, in weak-star, , in weak-star, .

Integrating (3.1) with respect to t, we have

Letting in (3.13), we get

and

On the other hand, from (3.2) and (3.3), we get in , in . Therefore u is a global weak solution of problem (1.1)-(1.3). □

In this section, we prove the main conclusion of this paper - the asymptotic behaviour of solution for problem (1.1)-(1.3).

Lemma 4.1Let (H₁) and (H₂) hold, , . Then, for the approximate solutionsof problem (1.1)-(1.3) constructed in the proof of Theorem 3.2, the following hold:

(i)

(ii) Furthermore, ifand, then for sufficiently largen, there exists asuch that

Proof (i) Multiplying (3.1) by and summing for s, we get (4.1).

(ii) From

it follows that there exists a such that

From (3.2), (3.3) and (4.3), it follows that for sufficiently large n. Hence from (3.5) we have

and

which gives

and

which together with for sufficiently large n gives

and

Hence, by Lemma 2.3, we have or . So, we have

 □

Theorem 4.2Let (H₁) and (H₂) hold, , . Assume that, . Then, for the global weak solutionugiven in Theorem 3.2, there exist positive constantsCandλsuch that

Proof Let be the approximate solutions of problem (1.1)-(1.3) in the proof of Theorem 3.2, then (3.4) holds. Multiplying (3.4) by (), we get

and

From (H₂), Lemma 2.2 and Lemma 4.1, we get

where

Hence we have

and

From

we get

and

which together with for sufficiently large n gives

and

From (4.8) and the Poincaré inequality , it follows that there exists a constant such that

From (4.5)-(4.10) it follows that there exists a such that

Choose α such that

Then from (4.11) we get

From this and the Gronwall inequality, we get

and

On the other hand, from (4.8) we get

Hence, there exists a such that

Let be the subsequence of in the proof of Theorem 3.2. Then from (4.13) and(4.12), we obtain

which gives (4.4), where , . □

The authors declare that they have no competing interests.

The work presented here was carried out in collaboration between all authors. RZ organised this paper and found the topic of this paper. He introduced this problem and suggested the methods and the outline of the proofs. JL finished the proof of the global existence and YN finished the long time behaviour part. SC discussed all the problems arising in the research and provided many good ideas for proving the problems. All authors have contributed to, seen and approved the manuscript.

We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11101102), Ph.D. Programs Foundation of the Ministry of Education of China (20102304120022), the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020), the Natural Science Foundation of Heilongjiang Province (A201014), Science and Technology Research Project of the Department of Education of Heilongjiang Province (12521401), Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities.

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